arXiv research

New solutions found for elliptic sinh-Gordon and sine-Gordon equations.

Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.

We solve integrable systems to describe the motion of Kaleidocycles.

Guichard's transformations generate Voss surfaces from sine-Gordon solutions.

In trying to provide explicit deformations of quadrics the starting point of our investigation is to use Bianchi's link between real deformations of totally real regions of real paraboloids and various totally real forms of the sine-Gordon equation coupled with Bianchi's simple observation that the vacuum soliton of th…

We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$ as solutions of sine-Gordon equation using Weierstrass's local parameters, which are characterized by two ramified points. Though the hyperelliptic solutions of the sine-Gord…

In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric conditio…

We present a unified method of construction of surfaces associated with Grassmannian sigma models, expressed in terms of an orthogonal projector. This description leads to compact formulae for structural equations of two-dimensional surfaces immersed in the su(N) algebra. In the special case of the CP^1 sigma model we …

Unified study of harmonic maps between pseudo-Riemannian surfaces.

Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, interesting first and higher flows of twi…

New harmonic maps to hyperbolic plane via Bäcklund transformation.

In this paper we continue investigation of the constant astigmatism equation z_{yy} + (1/z)_{xx} + 2 = 0. We newly interpret its solutions as describing spherical orthogonal equiareal patterns, with relevance to two-dimensional plasticity. We show how the classical Bianchi superposition principle for the sine-Gordon eq…

Recent results using inverse scattering techniques interpret every solution $φ(x,y)$ of the sine-Gordon equation as a non-linear superposition of solutions along the axes $x=0$ and $y=0$. Here we provide a geometric method of integration, as well as a geometric interpretation. Specifically, every weakly regular surface…

The class of traveling wave solutions of the sine-Gordon equation is known to be in 1-1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic…

By using Bäcklund transformation for the sine-Gordon equation, new periodic exact solutions of the constant astigmatism equation $z_{yy} + ({1}/{z})_{xx} + 2 = 0$ are generated from a seed which corresponds to Lipschitz surfaces of constant astigmatism.

The paper examines the neural tangent kernel for PINNs solving general PDEs and finds convergence conditions.

The paper characterizes surfaces in Heisenberg group with constant $p$-mean curvature.

We begin by considering several properties commonly (but not universally) possessed by Bäcklund transformations between hyperbolic Monge-Ampère equations: wavelike nature of the underlying equations, preservation of independent variables, quasilinearity of the transformation, and autonomy of the transformation. We show…

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the presence associated linear problems. It thus contains many important known examples…

We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation $z_{yy} + (1/z)_{xx} + 2 = 0$. The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter $λ= \pm1$. It is also a nonlocal symmetry…

Develops geometric framework for dissipative field equations.

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…

The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a 2-dimensional Riemannian metric of curvature equal to $-1$. The class of differe…

The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vo…

We introduce an algebraic formula producing infinitely many exact solutions of the constant astigmatism equation $z_{yy} + ({1}/{z})_{xx} + 2 = 0$ from a given seed. A construction of corresponding surfaces of constant astigmatism is then a matter of routine. As a special case, we consider multisoliton solutions of t…

Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel-Wu conjecture. The associated integrable nonlinear partial differential equation $z_{yy} + (1/z)_{xx} + 2 = 0$ possesses a zero curvature representation, a third-order symmetry, and a nonlocal transformatio…

We introduce two families of soliton hierarchies: the twisted hierarchies associated to symmetric spaces. The Lax pairs of these two hierarchies are Laurent polynomials in the spectral variable. Our constructions gives a hierarchy of commuting flows for the generalized sine-Gordon equation (GSGE), which is the Gauss-Co…

We show that the topological charge of the n-soliton solution of the sine-Gordon equation n is related to the genus g > 1 of a constant negative curvature compact surface described by this configuration. The relation is n=2(g-1), where n is even. The moduli space of complex dimension B(g)=3(g-1) corresponds precisely t…

Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…

Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …

In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…

This is an elementary and self--contained review of twistor theory as a geometric tool for solving non-linear differential equations. Solutions to soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or Sine-Gordon arise from holomorphic vector bundles over $T\CP^1$. A different framework is pro…

Two classification theorems for Willmore surfaces in S² × S².

Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.

Study finds negatively curved spheres in elliptic surfaces and their modifications.

The paper studies curve evolution using the PLR equation and its solutions.

In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in…

New method for analyzing elliptic and parabolic equations.

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

New topological obstructions found for elliptic and quasiregularly elliptic manifolds.

Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.

We consider the class of differential equations that describe pseudo-spherical surfaces of the form $u\_t=F(u,u\_x,u\_{xx})$ and $u\_{xt}=F(u, u\_x)$ given in Chern-Tenenblat \cite{ChernTenenblat} and Rabelo-Tenenblat \cite{RabeloTenenblat90}. We answer the following question: Given a pseudo-spherical surface determine…

This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using $KK$-theory.

The reduction problem of the chiral field equation on symmetric spaces is studied. It is shown that the symmetric chiral field has infinitely many local conservation laws. A recursive formula for these conservation laws is derived and the first associated integral of motion are given explicitly. Furthermore, the Zakhar…

Elliptic bouquets defined for spin manifolds with circular actions.

We give a parameterization of Alfred Gray's Elliptical Catenoid and Elliptical Hellicoid using Jacobi's elliptic functions. This parameterization avoids some problems present in the original depiction of these surfaces.

Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.

Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.